Problem: In the diagram, the smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is $6.$ What is the area of the shaded region?

[asy]
size(100);
import graph;
filldraw(Circle((0,0),2),mediumgray);
filldraw(Circle((-1,0),1),white);
filldraw(Circle((1,0),1),white);
[/asy]
Solution: Label the center of the larger circle $O$ and the points of contact between the larger circle and the smaller circles $A$ and $B.$ Draw the radius $OA$ of the larger circle.

[asy]
size(120);
import graph;
filldraw(Circle((0,0),2),mediumgray);
filldraw(Circle((-1,0),1),white);
filldraw(Circle((1,0),1),white);
draw((-2,0)--(0,0));

label("$A$",(-2,0),W); label("$O$",(0,0),E); label("$B$",(2,0),E);

[/asy]

Since the smaller circle and the larger circle touch at $A,$ the diameter through $A$ of the smaller circle lies along the diameter through $A$ of the larger circle. (This is because each diameter is perpendicular to the common tangent at the point of contact.)

Since $AO$ is a radius of the larger circle, it is a diameter of the smaller circle.

Since the radius of the larger circle is $6,$ the diameter of the smaller circle is $6,$ so the radius of the smaller circle on the left is $3.$

Similarly, we can draw a radius through $O$ and $B$ and deduce that the radius of the smaller circle on the right is also $3.$ The area of the shaded region equals the area of the larger circle minus the combined area of the two smaller circles. Thus, the area of the shaded region is $$6^2\pi - 3^2\pi - 3^2\pi = 36\pi - 9\pi - 9\pi = \boxed{18\pi}.$$